1. EECC756 - Shaaban #1 lec # 9 Spring2002 4-18-2002 Synchronous Iteration (Synchronous Parallelism) : –Barriers: Counter Barrier Implementation. Tree Barrier Implementation. Butterfly Connection Pattern Message-Passing Barrier. –Synchronous Iteration Program Example: Iterative Solution of Linear Equations. Dynamic Load BalancingDynamic Load Balancing –Centralized Dynamic Load Balancing. –Decentralized Dynamic Load Balancing: Distributed Work Pool Using Divide And Conquer. Distributed Work Pool With Local Queues In Slaves. Termination Detection for Decentralized Dynamic Load Balancing. –Program Example: Shortest Path Problem.

2. EECC756 - Shaaban #2 lec # 9 Spring2002 4-18-2002 Synchronous Iteration Iteration-based computation is a powerful method for solving numerical (and some non-numerical) problems. For numerical problems, a calculation is repeated and each time, a result is obtained which is used on the next execution. The process is repeated until the desired results are obtained. Though iterative methods are is sequential in nature, parallel implementation can be successfully employed when there are multiple independent instances of the iteration. In some cases this is part of the problem specification and sometimes one must rearrange the problem to obtain multiple independent instances. The term "synchronous iteration" is used to describe solving a problem by iteration where different tasks may be performing separate iterations but the iterations must be synchronized using point-to-point synchronization, barriers, or other synchronization mechanisms.

3. EECC756 - Shaaban #3 lec # 9 Spring2002 4-18-2002 Synchronous Iteration (Synchronous Parallelism) Each iteration composed of several processes that start together at beginning of iteration. Next iteration cannot begin until all processes have finished previous iteration. Using forall : for (j = 0; j < n; j++)/*for each synch. iteration */ forall (i = 0; i < N; i++) { /*N processes each using*/ body(i); /* specific value of i */ } or: for (j = 0; j < n; j++) {/*for each synchr. iteration */ i = myrank; /*find value of i to be used */ body(i); barrier(mygroup); }

4. EECC756 - Shaaban #4 lec # 9 Spring2002 4-18-2002 Barriers A synchronization mechanism applicable to shared-memory as well as message-passing, [pvm_barrier( ), MPI_barrier( )] where each process must wait until all members of a specific process group reach a specific reference point in their computation. Possible Implementations: –Using a counter (linear barrier). –Using individual point-to-point synchronization forming: A tree Butterfly connection pattern.

5. EECC756 - Shaaban #5 lec # 9 Spring2002 4-18-2002 Processes Reaching A Barrier At Different Times

6. EECC756 - Shaaban #6 lec # 9 Spring2002 4-18-2002 Centralized Counter Barrier Implementation Called linear barrier since access to centralized counter is serialized, thus O(n) time complexity.

7. EECC756 - Shaaban #7 lec # 9 Spring2002 4-18-2002 Message-Passing Counter Implementation of Barriers If the master process maintains the barrier counter: It counts the messages received from slave processes as they reach their barrier during arrival phase. Release slaves processes during departure phase after all the processes have arrived. for (i = 0; i <n; i++)/* count slaves as they reach their barrier */ recv(P any ); for (i = 0; i <n; i++)/* release slaves */ send(P i ); O(n) Time Complexity

8. EECC756 - Shaaban #8 lec # 9 Spring2002 4-18-2002 Tree Barrier Implementation 2 log n steps, time complexity O(log n)

9. EECC756 - Shaaban #9 lec # 9 Spring2002 4-18-2002 Tree Barrier Implementation Suppose 8 processes, P0, P1, P2, P3, P4, P5, P6, P7: Arrival phase: log8 = 3 stages –First stage: P1 sends message to P0; (when P1 reaches its barrier) P3 sends message to P2; (when P3 reaches its barrier) P5 sends message to P4; (when P5 reaches its barrier) P7 sends message to P6; (when P7 reaches its barrier) –Second stage: P2 sends message to P0; (P2 & P3 reached their barrier) P6 sends message to P4; (P6 & P7 reached their barrier) Third stage: P4 sends message to P0; (P4, P5, P6, & P7 reached barrier) P0 terminates arrival phase; (when P0 reaches barrier & received message from P4) Release phase also 3 stages with a reverse tree construction. Total number of steps: 2 log n = 2 log 8 = 6

10. EECC756 - Shaaban #10 lec # 9 Spring2002 4-18-2002 Butterfly Connection Pattern Message-Passing Barrier Butterfly pattern tree construction. Log n stages, thus O(log n) time complexity. Pairs of processes synchronize at each stage [two pairs of send( )/receive( )]. For 8 processes: First stage: P0  P1, P2  P3, P4  P5, P6  P7 Second stage: P0  P2, P1  P3, P4  P6, P5  P7 Third stage: P0  P4, P1  P5, P2  P6, P3  P7

11. EECC756 - Shaaban #11 lec # 9 Spring2002 4-18-2002 Message-Passing Local Synchronization Process P i-1 Process P i Process P i+1 recv(P i );send(P i-1 );recv(P i ); send(P i );send(P i+1 );send(P i ); recv(P i-1 ); recv(p i+1 );

12. EECC756 - Shaaban #12 lec # 9 Spring2002 4-18-2002 Synchronous Iteration Program Example: Iterative Solution of Linear Equations Given a system of n linear equations with n unknowns: a n-1,0 x 0 + a n-1,1 x 1 + a n-1,2 x 2...+ a n-1,n-1 x n-1 = b n-1. a 1,0 x 0 + a 1,1 x 1 + a 1,2 x 2... + a 1,n-1 x n-1 = b 1 a 0,0 x 0 + a 0,1 x 1 + a 0,2 x 2... + a 0,n-1 x n-1 = b 0 By rearranging the ith equation: a i,0 x 0 + a i,1 x 1 + a i,2 x 2... + a i,n-1 x n-1 = b i to: x i = (1/a i,i )[b i - (a i,0 x 0 + a i,1 x 1 + a i,2 x 2... a i,i-1 x i-1 + a i,i+1 x i+1 + a i,n-1 x n- 1 )] or This equation can be used as an iteration formula for each of the unknowns to obtain a better approximation. Jacobi Iteration: All the values of x are updated at once.

13. EECC756 - Shaaban #13 lec # 9 Spring2002 4-18-2002 Iterative Solution of Linear Equations Jacobi Iteration Sequential Code: Given the arrays a[][] and b[] holding the constants in the equations, x[] provided to hold the unknowns, and a fixed number of iterations, the code might look like: for (i = 0; i < n; i++) x[i] = b[i]; /* initialize unknowns */ for (iteration = 0; iteration < limit; iteration++) for (i = 0; i < n; i++) sum = 0; for (j = 0; j < n; j++) /* compute summation of a[][]x[] */ if (i != j) { sum = sum + a[i][j] * x[j]; new_x[i] = (b[i] - sum) / a[i][i]; /* Update unknown */ } for (i = 0; i < n; i++) /* update values */ x[i] = new_x[i]; }

14. EECC756 - Shaaban #14 lec # 9 Spring2002 4-18-2002 Iterative Solution of Linear Equations Jacobi Iteration Parallel Code: In the sequential code, the for loop is a natural "barrier" between iterations. In parallel code, we have to insert a specific barrier. Also all the newly computed values of the unknowns need to be broadcast to all the other processes. Process P i could be of the form: x[i] = b[i]; /* initialize values */ for (iteration = 0; iteration < limit; iteration++) { sum = -a[i][i] * x[i]; for (j = 1; j < n; j++) /* compute summation of a[][]x[] */ sum = sum + a[i][j] * x[j]; new_x[i] = (b[i] - sum) / a[i][i]; /* compute unknown */ broadcast_receive(&new_x[i]);/* broadcast values */ global_barrier(); /* wait for all processes */ } The broadcast routine, broadcast_receive(), sends the newly computed value of x[i] from process i to other processes and collects data broadcast from other processes.

15. EECC756 - Shaaban #15 lec # 9 Spring2002 4-18-2002 Partitioning Block allocation: –Allocate groups of n/p consecutive unknowns to processors in increasing order. Cyclic allocation: –Processors are allocated one unknown in order; –i.e., processor P0 is allocated x 0, x p, x 2p, …, x ((n/p)-1)p, processor P1 is allocated x 1, x p+1, x 2p+1, …, x ((n/p)-1)p+1, and so on. –Cyclic allocation has no particular advantage here (Indeed, may be disadvantageous because the indices of unknowns have to be computed in a more complex way).

16. EECC756 - Shaaban #16 lec # 9 Spring2002 4-18-2002 Jacobi Iteration: Analysis Sequential Time equals iteration time * number of iterations. O(n 2 ) for each iteration. Parallel execution time is the time of one processor each operating over n/p unknowns. Computation for  iterations: –Inner loop with n iterations, outer loop with n/p iterations –Inner loop: a multiplication and an addition. –Outer loop: a multiplication and a subtraction before inner loop and a subtraction and division after inner loop. t comp = n/p(2n + 4)  Time complexity O(n 2 /p) Communication: –Occurs at the end of each iteration, multiple broadcasts. – p broadcasts each of size n/p require t data to send each item t comm = p(t startup + (n/p)t data ) = (pt startup + nt data )  Overall Time: t p = (n/p(2n + 4)  pt startup + nt data ) 

17. EECC756 - Shaaban #17 lec # 9 Spring2002 4-18-2002 Effects of Computation And Communication in Jacobi Iteration For one iteration: t p = n/p(2n + 4)  pt startup + nt data Given: n = ? t startup = 10000 t data = 50 integer n/p Minimum execution time occurs when p = 16

18. EECC756 - Shaaban #18 lec # 9 Spring2002 4-18-2002 Other fully Synchronous Problems: Cellular Automata The problem space is divided into cells. Each cell can be in one of a finite number of states. Cells affected by their neighbors according to certain rules, and all cells are affected simultaneously in a “generation.” Rules re-applied in subsequent generations so that cells evolve, or change state, from generation to generation. Most famous cellular automata is the “Game of Life” devised by John Horton Conway, a Cambridge mathematician.

19. EECC756 - Shaaban #19 lec # 9 Spring2002 4-18-2002 The Game of Life Board game - theoretically infinite two-dimensional array of cells. Each cell can hold one “organism” and has eight neighboring cells, including those diagonally adjacent. Initially, some cells occupied. The following rules apply: –Every organism with two or three neighboring organisms survives for the next generation. –Every organism with four or more neighbors dies from overpopulation. –Every organism with one neighbor or none dies from isolation. –Each empty cell adjacent to exactly three occupied neighbors will give birth to an organism. These rules were derived by Conway “after a long period of experimentation.”

20. EECC756 - Shaaban #20 lec # 9 Spring2002 4-18-2002 Serious Applications for Cellular Automata Fluid/gas dynamics. The movement of fluids and gases around objects. Diffusion of gases. Biological growth. Airflow across an airplane wing. Erosion/movement of sand at a beach or riverbank.

21. EECC756 - Shaaban #21 lec # 9 Spring2002 4-18-2002 Dynamic Load Balancing To achieve best performance of a parallel computing system running a parallel problem, it’s essential to maximize processor utilization by distributing the computation load evenly or balancing the load among the available processors while minimizing overheads. Optimal static load balancing, mapping or scheduling, is an intractable NP-complete problem, except for specific problems on specific networks. Hence heuristics are usually used to select processors for processes. Even the best static mapping may not offer the best execution time due to changing conditions at runtime and the process mapping may need to done dynamically. The methods used for balancing the computational load dynamically among processors can be broadly classified as: 1. Centralized dynamic load balancing. 2. Decentralized dynamic load balancing.

22. EECC756 - Shaaban #22 lec # 9 Spring2002 4-18-2002 Processor Load Balance & Performance

23. EECC756 - Shaaban #23 lec # 9 Spring2002 4-18-2002 Centralized Dynamic Load Balancing A dvantage of centralized approach for computation termination: The master process terminates the computation when: 1. The task queue is empty, and 2. Every process has made a request for more tasks without any new tasks been generated.

24. EECC756 - Shaaban #24 lec # 9 Spring2002 4-18-2002 Decentralized Dynamic Load Balancing Distributed Work Pool Using Divide And Conquer

25. EECC756 - Shaaban #25 lec # 9 Spring2002 4-18-2002 Decentralized Dynamic Load Balancing Distributed Work Pool With Local Queues In Slaves Termination Conditions for Decentralized Dynamic Load Balancing: In general, termination at time t requires two conditions to be satisfied: 1. Application-specific local termination conditions exist throughout the collection of processes, at time t, and 2. There are no messages in transit between processes at time t. Tasks could be transferred by: 1. Receiver-initiated method. 2. Sender-initiated method.

26. EECC756 - Shaaban #26 lec # 9 Spring2002 4-18-2002 Termination Detection for Decentralized Dynamic Load Balancing Ring Termination Algorithm: –Processes organized in ring structure. –When P 0 terminated it generates a token to P 1. –When Pi receives the token and has already terminated, it passes the token to P i+1. P n-1 passes the token to P 0 –When P 0 receives the token it knows that all processes in ring have terminated. A message can be sent to all processes informing them of global termination if needed.

27. EECC756 - Shaaban #27 lec # 9 Spring2002 4-18-2002 Program Example: Shortest Path Algorithm Given a set of interconnected vertices or nodes where the links between nodes have associated weights or “distances”, find the path from one specific node to another specific node that has the smallest accumulated weights. One instance of the above problem below: –“Find the best way to climb a mountain given a terrain map.” Mountain Terrain Map Corresponding Graph

28. EECC756 - Shaaban #28 lec # 9 Spring2002 4-18-2002 Representation of Sample Problem Graph Problem Graph

29. EECC756 - Shaaban #29 lec # 9 Spring2002 4-18-2002 Moore’s Single-source Shortest-path Algorithm Starting with the source, the basic algorithm implemented when vertex i is being considered is as follows. –Find the distance to vertex j through vertex i and compare with the current distnce directly to vertex j. –Change the minimum distance if the distance through vertex j is shorter. If d i is the distance to vertex i, and w i j is the weight of the link from vertex i to vertexj, we have: d j = min(d j, d i +w i j ) The code could be of the form: newdist_j = dist[i]+w[i][j]; if(newdist_j < dist[j]) dist[j] = newdist_j; When a new distance is found to vertex j, vertex j is added to the queue (if not already in the queue), which will cause this vertex to be examined again.

30. EECC756 - Shaaban #30 lec # 9 Spring2002 4-18-2002 Steps of Moore’s Algorithm for Example Graph Stages in searching the graph: –Initial values –Each edge from vertex A is examined starting with B –Once a new vertex, B, is placed in the vertex queue, the task of searching around vertex B begins. The weight to vertex B is 10, which will provide the first (and actually the only distance) to vertex B. Both data structures, vertex_queue and dist[] are updated. The distances through vertex B to the vertices are dist[F]=10+51=61, dist[E]=10+24=34, dist[D]=10+13=23, and dist[C]= 10+8=18. Since all were new distances, all the vertices are added to the queue (except F) Vertex F need not to be added because it is the destination with no outgoing edges and requires no processing.

31. EECC756 - Shaaban #31 lec # 9 Spring2002 4-18-2002 Starting with vertex E: –It has one link to vertex F with the weight of 17, the distance to vertex F through vertex E is dist[E]+17= 34+17= 51 which is less than the current distance to vertex F and replaces this distance. Next is vertex D: –There is one link to vertex E with the weight of 9 giving the distance to vertex E through vertex D of dist[D]+ 9= 23+9 = 32 which is less than the current distance to vertex E and replaces this distance. –Vertex E is added to the queue. Steps of Moore’s Algorithm for Example Graph

32. EECC756 - Shaaban #32 lec # 9 Spring2002 4-18-2002 Steps of Moore’s Algorithm for Example Graph Next is vertex C: – We have one link to vertex D with the weight of 14. –Hence the (current) distance to vertex D through vertex C of dist[C]+14= 18+14=32. This is greater than the current distance to vertex D, dist[D], of 23, so 23 is left stored. Next is vertex E (again): –There is one link to vertex F with the weight of 17 giving the distance to vertex F through vertex E of dist[E]+17= 32+17=49 which is less than the current distance to vertex F and replaces this distance, as shown below: There are no more vertices to consider and we have the minimum distance from vertex A to each of the other vertices, including the destination vertex, F. Usually the actual path is also required in addition to the distance and the path needs to be stored as the distances are recorded. The path in our case is ABDE F.

33. EECC756 - Shaaban #33 lec # 9 Spring2002 4-18-2002 Moore’s Single-source Shortest-path Algorithm Sequential Code: The specific details of maintaining the vertex queue are omitted. Let next_vertex() return the next vertex from the vertex queue or no_vertex if none, and let next_edge() return the next link around a vertex to be considered. (Either an adjacency matrix or an adjacency list would be used to implement next_edge()). The sequential code could be of the form: while ((i=next_vertex())!=no_vertex) /* while there is a vertex */ while (j=next_edge(vertex)!=no_edge) { /* get next edge around vertex */ newdist_j=dist[i] + w[i][j]; if (newdist_j < dist[j]) { dist[j]=newdist_j; append_gueue(j); /* add vertex to queue if not there */ } } /* no more vertices to consider */

34. EECC756 - Shaaban #34 lec # 9 Spring2002 4-18-2002 Moore’s Single-source Shortest-path Algorithm Parallel Implementation, Centralized Work Pool The code could be of the form: Master recv(any, P i ); /* request for task from process P i */ if ((i= next_edge()!= no_edge) send(P i, i, dist[i]); /* send next vertex, and. /* current distance to vertex */ recv(P j, j, dist[j]); /* receive new distances */ append_gueue(j); /* append vertex to queue */. Slave (process i) send(P master, P i ); /* send a request for task */ recv(P master, i, d); /* get vertex number and distance */ while (j=next_edge(vertex)!= no_edge) { /* get next link around vertex */ newdist_j = d + w[i][j]; if (newdist_j < dist[j]) { dist[j]=newdist_j; send(P master, j, dist[j]); /* send back updated distance */ } } /* no more vertices to consider */

35. EECC756 - Shaaban #35 lec # 9 Spring2002 4-18-2002 Moore’s Single-source Shortest-path Algorithm Parallel Implementation, Decentralized Work Pool The code could be of the form: Master if ((i = next_vertex()!= no_vertex) send(P i, "start"); /* start up slave process i */. Slave (process i). if (recv(P j, msgtag = 1)) /* asking for distance */ send(P j, msgtag = 2, dist[i]); /* sending current distance */. if (nrecv(P master ) { /* if start-up message */ while (j=next_edge(vertex)!=no_edge) { /* get next link around vertex */ newdist_j = dist[i] + w[j]; send(P j, msgtag=1); /* Give me the distance */ recv(P i, msgtag = 2, dist[j]); /* Thank you */ if (newdist_j > dist[j]) { dist[j] = newdist_j; send(P j, msgtag=3, dist[j]); /* send updated distance to proc. j */ } where w[j] hold the weight for link from vertex i to vertex j.

36. EECC756 - Shaaban #36 lec # 9 Spring2002 4-18-2002 Moore’s Single-source Shortest-path Algorithm Distributed Graph Search